= Solution
At resonance, retain the signed coupling $\Omega=\gamma B_x/2$. In the <interaction picture>, the given solution reduces to $c_1=\cos(\Omega t/2)$ and $c_2=i\sin(\Omega t/2)$. Returning to the <Schrödinger picture> gives
$$
|\psi(t)\rangle=e^{i\omega_zt/2}\cos(\Omega t/2)|\uparrow\rangle
+i e^{-i\omega_zt/2}\sin(\Omega t/2)|\downarrow\rangle.
$$
For a normalized spinor $(a,b)^T$, the <Pauli matrices> give $\langle\boldsymbol\sigma\rangle=(2\operatorname{Re}a^*b,2\operatorname{Im}a^*b,|a|^2-|b|^2)$. Therefore the <spin> <expectation value> within the <rotating-wave approximation> is
$$
\boxed{\langle\mathbf S\rangle_t=\frac\hbar2
\begin{pmatrix}
\sin(\Omega t)\sin(\omega_zt)\\
\sin(\Omega t)\cos(\omega_zt)\\
\cos(\Omega t)
\end{pmatrix}.}
$$
Its magnitude is $\hbar/2$: the <Bloch vector> stays on the unit sphere. In a frame rotating about $z$ at angular velocity $-\omega_z$, it makes a uniform rotation in the $yz$ plane. For $|B_x|\ll|B_z|$, this gives slow <Rabi oscillations> between the poles, accompanied by rapid laboratory <Larmor precession> about $z$.
If the same formula is formally used for $|B_z|\ll|B_x|$, it predicts fast meridional rotations whose plane turns slowly about $z$. However, the original PDF really does ask for this opposite limit, and it lies outside the assumption used in part (b). That formal description is not a controlled prediction for the stated linearly driven <Hamiltonian operator>.
A valid strong-drive description starts from the exact equation for the <Bloch vector> $\mathbf s=2\langle\mathbf S\rangle/\hbar$:
$$
\frac{d\mathbf s}{dt}=\gamma\mathbf s\times\bigl(B_x\cos(\omega t),0,B_z\bigr).
$$
For times $|\gamma B_z|t\ll1$, neglecting the weak bias gives a controlled leading approximation, since the time integral of the omitted <Hamiltonian operator> has small norm. The remaining field has a fixed $x$ axis, so its generators commute at different times and
$$
\mathbf s(t)\simeq\bigl(0,\sin\Theta(t),\cos\Theta(t)\bigr),
\qquad\Theta(t)=\frac{\gamma B_x}{\omega}\sin(\omega t).
$$
At the nominal resonance $\omega=\gamma B_z$ and on the early spin-rotation scale $t\sim1/|\gamma B_x|$, this becomes $\Theta\simeq\gamma B_xt$. The initial rotation rate is consequently $\gamma B_x$, twice the rate predicted by extrapolating the <rotating-wave approximation>. For longer times the exact differential equation must be retained. The following original calculation illustrates both regimes rather than asserting an exact strong-drive Rabi formula.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2021-q4-spin-motion.png]
{title=Laboratory spin trajectories for weak and strong transverse driving}
{description=Original solutions of the full time-dependent spin equation at nominal resonance, compared with the <rotating-wave approximation>. Top panels show the <Bloch vector> trajectory (the first half-cycle for weak driving); bottom panels compare its $z$ component. Time is $\tau=\gamma B_zt$. The weak-drive case agrees closely over a Rabi cycle; the strong-drive case demonstrates why the approximation cannot be extrapolated to $|B_x|\gg|B_z|$.}
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